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The projective general linear group \({\mathrm {PGL}}(2,2^m)\) and linear codes of length \(2^m+1\)

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Abstract

Let \(q=2^m\). The projective general linear group \({\mathrm {PGL}}(2,q)\) acts as a 3-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over \({\mathrm {GF}}(2^h)\) that are invariant under \({\mathrm {PGL}}(2,q)\) are trivial codes: the repetition code, the whole space \({\mathrm {GF}}(2^h)^{2^m+1}\), and their dual codes. As an application of this result, the 2-ranks of the (0,1)-incidence matrices of all \(3-(q+1,k,\lambda )\) designs that are invariant under \({\mathrm {PGL}}(2,q)\) are determined. The second objective is to present two infinite families of cyclic codes over \({\mathrm {GF}}(2^m)\) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under \({\mathrm {PGL}}(2,q)\), therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters \([q+1,q-3,4]_q\), where \(q=2^m\), and \(m\ge 4\) is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3-\((q+1,4,2)\) design. A code from the second family has parameters \([q+1,4,q-4]_q\), \(q=2^m\), \(m\ge 4\) even, and the minimum weight codewords support a 3-\((q +1,q-4,(q-4)(q-5)(q-6)/60)\) design, whose complementary 3-\((q +1, 5, 1)\) design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over \({\mathrm {GF}}(q)\) that can support a 3-\((q +1,q-4,(q-4)(q-5)(q-6)/60)\) design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound.

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References

  1. Assmus Jr. E.F., Mattson Jr. H.F.: New 5-designs. J. Comb. Theory 6, 122–151 (1969).

    Article  MathSciNet  Google Scholar 

  2. Beth T., Jungnickel D., Lenz H.: Design Theory. Cambridge University Press, Cambridge (1999).

    Book  Google Scholar 

  3. Bosma W., Cannon J.: Handbook of Magma Functions. University of Sydney, School of Mathematics and Statistics, Sydney (1999).

    Google Scholar 

  4. Colbourn C.J., Dinitz J.F.: Handbook of Combinatorial Designs, 2nd edn. Chapman & Hall/CRC, Boca Raton (2007).

    MATH  Google Scholar 

  5. Dickson L.E.: Linear Groups: With an Exposition of the Galois Field Theory. Teubner, Leipzig (1901).

    MATH  Google Scholar 

  6. Delsarte P.: On subfield subcodes of modified Reed-Solomon codes. IEEE Trans. Inform. Theory 21(5), 575–576 (1975).

    Article  MathSciNet  Google Scholar 

  7. Ding C.: Designs from Linear Codes. World Scientific, Singapore (2018).

    Book  Google Scholar 

  8. Ding C., Tang C.: Infinite families of near MDS codes holding \(t\)-designs. IEEE Trans. Inform. Theory 66(9), 5419–5428 (2020).

    Article  MathSciNet  Google Scholar 

  9. Du X., Wang R., Fan C.: Infinite families of \(2\)-designs from a class of cyclic codes. J. Comb. Des. 28(3), 157–170 (2020).

    Article  MathSciNet  Google Scholar 

  10. Giorgetti M., Previtali A.: Galois invariance, trace codes and subfield subcodes. Finite Fields Appl. 16(2), 96–99 (2010).

    Article  MathSciNet  Google Scholar 

  11. Huber M.: The classification of flag-transitive Steiner 3-designs. Adv. Geom. 5(2), 195–221 (2005).

    Article  MathSciNet  Google Scholar 

  12. Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).

    Book  Google Scholar 

  13. Hughes D.R.: On \(t\)-designs and groups. Am. J. Math. 87(4), 761–778 (1965).

    Article  MathSciNet  Google Scholar 

  14. Jungnickel D., Magliveras S.S., Tonchev V.D., Wassermann A.: The classification of Steiner triple systems on 27 points with 3-rank 24. Des. Codes Cryptogr. 87, 831–839 (2019).

    Article  MathSciNet  Google Scholar 

  15. Jungnickel D., Tonchev V.D.: New invariants for incidence structures. Des. Codes Cryptogr. 68, 163–177 (2013).

    Article  MathSciNet  Google Scholar 

  16. Jungnickel D., Tonchev V.D.: Counting Steiner triple systems with classical parameters and prescribed rank. J. Comb. Theory Ser. A 162, 10–33 (2019).

    Article  MathSciNet  Google Scholar 

  17. Passman D.S.: Permutation Groups. Benjamin, New York (1968).

    MATH  Google Scholar 

  18. Shi M., Xu L., Krotov D.S.: The number of the non-full-rank Steiner triple systems. J. Comb. Des. 27(10), 571–585 (2019).

    Article  MathSciNet  Google Scholar 

  19. Tang C.: Infinite families of \(3\)-designs from APN functions. J. Comb. Des. 28(2), 97–117 (2020).

    Article  MathSciNet  Google Scholar 

  20. Tang C., Ding C.: An infinite family of linear codes supporting \(4\)-designs. IEEE Trans. Inform. Theory 67(1), 244–254 (2021).

    Article  MathSciNet  Google Scholar 

  21. Tonchev V.D.: Linear perfect codes and a characterization of the classical designs. Des. Codes Cryptogr. 17, 121–128 (1999).

    Article  MathSciNet  Google Scholar 

  22. Tonchev V.D.: Codes. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 677–701. CRC Press, New York (2007).

    Google Scholar 

  23. Witt E.: Über Steinersche Systeme. Abh. Math. Sem. Hamburg 12, 265–275 (1938).

    Article  Google Scholar 

  24. Xiang C., Ling X., Wang Q.: Combinatorial \(t\)-designs from quadratic functions. Des. Codes Cryptogr. 88(3), 553–565 (2020).

    Article  MathSciNet  Google Scholar 

  25. Zinoviev D.V.: The number of Steiner triple systems \(S(2^m - 1, 3, 2)\) of rank \(2^m -m+2\) over \(\mathbb{F}_2\). Discrete Math. 339, 2727–2736 (2016).

    Article  MathSciNet  Google Scholar 

  26. Zinoviev V.A., Zinoviev D.V.: Steiner triple systems \(S(2^m - 1, 3, 2)\) of rank \(2^m -m+1\) over \(\mathbb{F}_2\). Probl. Inf. Transm. 48, 102–126 (2012).

    Article  MathSciNet  Google Scholar 

  27. Zinoviev V.A., Zinoviev D.V.: Structure of Steiner triple systems \(S(2^m - 1, 3, 2)\) of rank \(2^m -m+2\) over \(\mathbb{F}_2\). Probl. Inf. Transm. 49, 232–248 (2013).

    Article  MathSciNet  Google Scholar 

  28. Zinoviev V.A., Zinoviev D.V.: Remark on Steiner triple systems \( S(2^m - 1, 3, 2)\) of rank \(2^m -m+1\) over \( {{\mathbb{F}}_2}\) published in Probl. Peredachi Inf., 2012, no. 2. Probl. Inf. Transm. 49, 107–111 (2013).

    Google Scholar 

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Acknowledgements

The authors are grateful to the reviewers and the Editors for their comments and suggestions that improved the presentation of this paper.

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Correspondence to Chunming Tang.

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Communicated by J. W. P. Hirschfeld.

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C. Ding’s research was supported by the Hong Kong Research Grants Council, Proj. No. 16300418. C. Tang’s research was supported by The National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds).

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Ding, C., Tang, C. & Tonchev, V.D. The projective general linear group \({\mathrm {PGL}}(2,2^m)\) and linear codes of length \(2^m+1\). Des. Codes Cryptogr. 89, 1713–1734 (2021). https://doi.org/10.1007/s10623-021-00888-2

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